I saw that you mentioned that Alex has some health issues requiring medication and specific diet. I wonder if part of the reason he hasn't been looking for jobs could be side effects of his meds or a side effect/comorbidity of one of his health issues. You don't say whether it's a physical or mental health issue, but if it's bad enough he may qualify for some type of disability/benefits. Of course it's not as good $wise as a full or even part time job may be but it could be a start on the correct path for him to become a functioning adult.

I say this only because for years my sibling seemed to struggle with similar problems and it took us about 3 years from the start to find out that they had severe depression with agoraphobia on top of a recurring pilonidal cyst which made it difficult to sit for long periods of time or do any kind of physical labor. Some of the side effects of medications or even undiagnosed health issues can make it difficult to work, and some things (like my sibling's cyst) are so embarrassing people are hesitant to seek help (also like my sibling. Who wants to admit they have a cyst on their butt?).

I think your compromise is good - Alex should definitely be doing something and have something to show regarding getting his life in order - but I think you and Nathan need to sit down and have a discussion with Alex about what that might look like. Maybe it's enrolling in community college classes or picking up a part time job (depending on where you live it's likely he'll be hired for part time work only and need 2 or 3 jobs to reach full time hours and that can be super disheartening for young adults) or if his health issues are bad enough/being unaddressed at the moment maybe it's taking care of that stuff and applying for disability/benefits/seeing a case worker/more intensive health stuff to get his issues under control.

Also - help where you can, don't stretch yourself thin. If you and Nathan absolutely don't want to take Alex in, help him find a shelter or youth hostel to stay at to give yourselves peace of mind so you're not wracked with guilt over this. There's also programs in some cities that provide housing for young adults in exchange for work/volunteering which you can also look into. Some of the commenters are absolutely condemning your sibling here for not being an adult and freeloading, but without more information on what's actually going on in Alex's life, that may not be the case. It's possible that your sibling just needs to be empowered and shown he has the capability TO get things done before he feels able to actually DO it, you know?